Vibrations of a cantilever without damping are described by D. Landau and E. M. Lifshitz (Theory of Elasticity, 3rd ed., vol. 7. Oxford: Butterworth-Heinemann, 1986).
Micro-machined cantilevers are excellent sensors as they are extremely sensitive and miniature, mass produced. and low cost. They operate by detecting changes either in resonant frequency, amplitude, Q-factor or in deflection caused by mass loading, surface stress variation, or any other changes of the cantilever's environment.
Sensors for measuring physical parameters with a vibrating element are known (WO2005043126, Rasmussen et al in Ultramicroscopy 97, 2003 371-376), for example, such sensors may measure pressure. In these devices, a physical parameter is measured without compensation for changes in other parameters in the environment, such as temperature.
As these sensors often show a temperature dependency, various solutions to compensate the output of the sensor for temperature variations have been proposed (EP 0 371 592 B1, U.S. Pat. Nos. 4,554,927, 5,379,639, 5,458,000, 5,511,427). These patents use two different (vibrating) means to measure pressure and temperature.
For measuring both temperature and pressure, J. Mertens et al (Ultramicroscopy 97, 2003, 119-126) and R. Sandberg et al (J. Micromech. Microeng. 15, 2005, 1454-1458) proposed one single microcantilever. In both cases a piezoelectric transducer actuates the cantilever. Resonant frequencies were determined by cantilever deflection measurements from the deflection of a laser beam reflecting off the cantilever top surface. Extra optical means are required for measuring the cantilever deflection by this method, and it is not straightforward.
Also known in the field is the use of piezoelectric materials, both for electrical actuation and electrical sensing. Here, a change in the resonance frequency allows defining a change in a parameter, such as a change in viscosity of the surrounding medium, or a change in mass load of the cantilever (WO2005/043126).
U.S. Pat. No. 4,144,747 and Zheyao Wang et al in Meas. Sci. Technol. 11, 1565-1569 (2000) mention the use of two resonant modes of a quartz resonator to measure a force, thereby compensating for the temperature.
However, there is a need to measure more than one parameter using only one piezoelectric bimorph cantilever in a gaseous or liquid environment so as to compensate for changes in other parameters. It is also desirable to make such a sensor very small and highly sensitive.